A100405 -id:A100405 - cp's OEIS frontend

A160974 Number of partitions of n where every part appears at least 4 times.

1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 4, 4, 7, 5, 8, 7, 13, 10, 13, 12, 21, 18, 22, 21, 34, 29, 40, 36, 55, 48, 63, 64, 88, 79, 100, 99, 139, 125, 160, 155, 207, 199, 241, 241, 314, 302, 369, 366, 466, 454, 550, 557, 690, 679, 807, 821, 1016, 1001, 1180, 1207, 1460, 1466, 1708

Offset: 0

R. H. Hardin, Jun 01 2009

nonn

			a(12) = 4 because we have 3333, 2^6, 22221111, and 1^(12). - _Emeric Deutsch_, Jun 24 2009

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..967 from R. H. Hardin)

Cf. A007690, A100405, A160975-A160990.

g := product(1+x^(4*j)/(1-x^j), j = 1..30): gser := series(g, x = 0, 85): seq(coeff(gser, x, n), n = 0..66); # Emeric Deutsch, Jun 24 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=[0, $4..iquo(n, i)])))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 02 2017

nmax = 100; CoefficientList[Series[Product[1 + x^(4*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015; offset adapted by Georg Fischer, Sep 18 2020 *)

G.f.: Product_{j>=1} (1+x^(4*j)/(1-x^j)). - Emeric Deutsch, Jun 24 2009

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-4*x)) dx = -0.903005550655893892139378653023287247062261773608753265529... . - Vaclav Kotesovec, Jan 05 2016

Initial terms changed to match b-file. - N. J. A. Sloane, Aug 31 2009
Maple program fixed by Vaclav Kotesovec, Nov 28 2015
a(0)=1 prepended by Alois P. Heinz, Oct 02 2017

A353502 Numbers with all prime indices and exponents > 2.

Original entry on oeis.org

1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911

Offset: 1

Gus Wiseman, May 16 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The initial terms together with their prime indices:
       1: {}
     125: {3,3,3}
     343: {4,4,4}
     625: {3,3,3,3}
    1331: {5,5,5}
    2197: {6,6,6}
    2401: {4,4,4,4}
    3125: {3,3,3,3,3}
    4913: {7,7,7}
    6859: {8,8,8}
   12167: {9,9,9}
   14641: {5,5,5,5}
   15625: {3,3,3,3,3,3}
   16807: {4,4,4,4,4}
   24389: {10,10,10}
   28561: {6,6,6,6}
   29791: {11,11,11}
   42875: {3,3,3,4,4,4}

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Amiram Eldar)

The version for only parts is A007310, counted by A008483.
The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
The version for only multiplicities is A036966, counted by A100405.
The version for indices and exponents prime (instead of > 2) is:
- listed by A346068
- counted by A351982
- only exponents: A056166, counted by A055923
- only parts: A076610, counted by A000607
The version for > 1 instead of > 2 is A062739, counted by A339222.
The version for compositions is counted by A353428, see A078012, A353400.
The partitions with these Heinz numbers are counted by A353501.
A000726 counts partitions with multiplicities <= 2, compositions A128695.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.

Select[Range[10000],#==1||!MemberQ[FactorInteger[#],{?(#<5&),}|{,?(#<3&)}]&]

Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)*A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022

A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Alois P. Heinz, Table of n, a(n) for n = 0..4000

The first condition only is A023360, partitions A000607.
For partitions we have A351982, only run-lens A100405, only parts A008483.
The second condition only is A353401, partitions A055923.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A052284 counts compositions into nonprimes, partitions A002095.
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A329738 counts uniform compositions, partitions A047966.
Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]

a(26)-a(56) from Alois P. Heinz, May 18 2022

A353508 Number of integer compositions of n with no ones or runs of length 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 8, 2, 11, 4, 21, 5, 37, 12, 57, 25, 104, 38, 177, 79, 292, 149, 513, 251, 876, 482, 1478, 871, 2562, 1533, 4387, 2815, 7473, 5036, 12908, 8935, 22135, 16085, 37940, 28611, 65422, 50731, 112459, 90408, 193386, 160119, 333513

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(14) = 11 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  .  .  .  22  .  33   .  44    333  55     .  66      22333  77
                     222     2222       2233      444     33322  2255
                                        3322      2244           3344
                                        22222     3333           4433
                                                  4422           5522
                                                  22233          22244
                                                  33222          44222
                                                  222222         222233
                                                                 223322
                                                                 332222
                                                                 2222222

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The version for partitions is A339222.
Compositions counted by their run-lengths:
- For run-lengths <= 1 we have A003242, ranked by A333489.
- For run-lengths = 2 we have A003242 aerated.
- For run-lengths > 1 we have A114901, ranked by A353427.
- For run-lengths <= 2 we have A128695 matching A335464.
- For run-lengths > 2 we have A353400, partitions A100405.
- For run-lengths all prime we have A353401.
- For run-lengths and parts > 2 we have A353428.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A261983 counts non-anti-run compositions.
A274174 counts compositions with equal parts contiguous.
Cf. A005811, A078012, A238279, A329739, A333755, A351013, A353390.

b:= proc(n,h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&!MemberQ[Length/@Split[#],1]&]],{n,0,15}]

a(41)-a(52) from Alois P. Heinz, May 17 2022

A160975 Number of partitions of n where every part appears at least 5 times.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 7, 7, 7, 8, 11, 12, 12, 14, 15, 16, 23, 20, 24, 26, 29, 36, 40, 40, 46, 50, 63, 63, 76, 76, 87, 103, 108, 117, 135, 140, 167, 173, 191, 205, 235, 257, 278, 300, 327, 354, 413, 424, 469, 511, 555, 616, 673, 711, 783, 849, 947

Offset: 0

R. H. Hardin, Jun 01 2009

nonn

			a(15) = 3 because we have 33333, 2222211111, and 1^(15). - _Emeric Deutsch_, Jun 28 2009

Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 1..1000 from R. H. Hardin)

Cf. A007690, A100405, A160974, A160976-A160990.

g := product(1+x^(5*j)/(1-x^j), j = 1..20): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0..75); # Emeric Deutsch, Jun 28 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1), j=5..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 06 2024

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(5*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2015 *)

G.f.: Product_{j>=1} (1+x^(5*j)/(1-x^j)). - Emeric Deutsch, Jun 28 2009

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-5*x)) dx = -0.990807844177842472956484606320623872921836802804155824925... . - Vaclav Kotesovec, Jan 05 2016

Initial terms changed to match b-file. - N. J. A. Sloane, Aug 31 2009
Maple program fixed by Vaclav Kotesovec, Nov 28 2015
a(0)=1 prepended by Seiichi Manyama, Feb 06 2024

A161294 Number of partitions of n into numbers not divisible by 4 where every part appears at least 3 times.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 10, 8, 9, 15, 13, 16, 22, 20, 24, 33, 32, 36, 47, 48, 53, 71, 68, 77, 100, 99, 112, 140, 138, 158, 194, 199, 219, 268, 275, 305, 369, 377, 416, 501, 514, 572, 671, 693, 768, 898, 935, 1028, 1189, 1245, 1364, 1576, 1642, 1798, 2063

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

			a(13)=5 because we have (3^3)(1^4), (2^5)(1^3), (2^4)(1^5), (2^3)(1^7), and 1^(13). - _Emeric Deutsch_, Jun 21 2009

R. H. Hardin, Table of n, a(n) for n = 1..1000

Cf. A100405.

g := -1+(product(1+x^(3*j)/(1-x^j), j = 1 .. 40))/(product(1+x^(12*j)/(1-x^(4*j)), j = 1 .. 40)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 2 .. 68); # Emeric Deutsch, Jun 21 2009

(* Closed form for the constant c: *) N[Pi^2/3 + 1/2*(Log[-1/3 - 1/6*(1 + I*Sqrt[3])*(25/2 - (3*Sqrt[69])/2)^(1/3) - 1/6*(1 - I*Sqrt[3])*(1/2*(25 + 3*Sqrt[69]))^(1/3)]^2 + Log[-1/3 - 1/6*(1 - I*Sqrt[3])*(25/2 - (3*Sqrt[69])/2)^(1/3) - 1/6*(1 + I*Sqrt[3])*((1/2)*(25 + 3*Sqrt[69]))^(1/3)]^2 + 2*(-PolyLog[2, 1/3*(1 - (2/(25 - 3*Sqrt[69]))^(1/3) - (1/2*(25 - 3*Sqrt[69]))^(1/3))] + PolyLog[2, ((1 + I*Sqrt[3])*(1/2*(9 - Sqrt[69]))^(1/3))/(2*3^(2/3)) + (1 - I*Sqrt[3])/(2^(2/3)*(3*(9 - Sqrt[69]))^(1/3))] + PolyLog[2, ((1 - I*Sqrt[3])*(1/2*(9 - Sqrt[69]))^(1/3)) / (2*3^(2/3)) + (1 + I*Sqrt[3])/(2^(2/3)*(3*(9 - Sqrt[69]))^(1/3))])), 100] // Chop (* Vaclav Kotesovec, Jun 15 2025 *)

G.f.: -1 + (Product_{j>=1} (1 + x^(3*j)/(1-x^j)))/Product_{j>=1} (1 + x^(12*j)/(1-x^(4*j))). - Emeric Deutsch, Jun 21 2009

a(n) ~ (6*c + Pi^2)^(1/4) * exp(sqrt((6*c + Pi^2)*n/2)) / (2^(11/4) * sqrt(Pi) * n^(3/4)), where c = Integral_{x=0..oo} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... - Vaclav Kotesovec, Jun 15 2025

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) partitions for selected n (A = 10):
  n=9:   n=12:   n=21:      n=24:       n=30:
------------------------------------------------------
  (333)  (444)   (777)      (888)       (AAA)
         (3333)  (444333)   (6666)      (66666)
                 (3333333)  (444444)    (555555)
                            (555333)    (666444)
                            (4443333)   (777333)
                            (33333333)  (6663333)
                                        (55533333)
                                        (444333333)
                                        (3333333333)

The version for only parts > 2 is A008483.
The version for only multiplicities > 2 is A100405.
The version for parts and multiplicities > 1 is A339222, ranked by A062739.
For prime parts and multiplicities we have A351982, compositions A353429.
The version for compositions is A353428 (partial A078012, A353400).
These partitions are ranked by A353502.
A000726 counts partitions with all mults <= 2, compositions A128695.
A004250 counts partitions with some part > 2, compositions A008466.
A137200 counts compositions with all parts and run-lengths <= 2.
Cf. A003242, A047966, A055923, A098859, A114901, A325534, A333755, A335464.

Table[Length[Select[IntegerPartitions[n],Min@@#>2&&Min@@Length/@Split[#]>2&]],{n,0,30}]

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Gus Wiseman, May 23 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

A161039 Number of partitions of n into odd numbers where every part appears at least 3 times.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 5, 3, 3, 6, 5, 6, 8, 6, 7, 11, 10, 9, 14, 13, 13, 19, 16, 18, 25, 22, 25, 32, 29, 31, 42, 41, 41, 53, 51, 54, 69, 64, 69, 88, 83, 89, 109, 105, 112, 136, 134, 141, 170, 166, 177, 215, 207, 219, 262, 260, 276, 320, 320, 341, 397, 397, 417, 485

Offset: 1

R. H. Hardin Jun 02 2009

nonn

			a(15)=5 because we have 333, (2^6)(1^3), (2^5)(1^5), (2^4)(1^7), and (2^3)(1^9).

R. H. Hardin and Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. H. Hardin)

Cf. A100405.

g := product(1+x^(3*(2*j-1))/(1-x^(2*j-1)), j = 1 .. 20): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 1 .. 72); # Emeric Deutsch, Jun 26 2009

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(6*k-3) / (1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

G.f.: Product_{j>=1} (1 + x^(6j-3)/(1-x^(2j-1))). - Emeric Deutsch, Jun 26 2009

a(n) ~ (6*c + Pi^2)^(1/4) * exp(sqrt((6*c + Pi^2)*n/3)) / (4*3^(1/4)*sqrt(Pi) * n^(3/4)), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... . - Vaclav Kotesovec, Jan 05 2016

Minor edits by Vaclav Kotesovec, Jan 02 2016

cp's OEIS Frontend

A160974 Number of partitions of n where every part appears at least 4 times.

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A353502 Numbers with all prime indices and exponents > 2.

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A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

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A353508 Number of integer compositions of n with no ones or runs of length 1.

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A160975 Number of partitions of n where every part appears at least 5 times.

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A161294 Number of partitions of n into numbers not divisible by 4 where every part appears at least 3 times.

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A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

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A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

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